DefinitionDefinition: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Binary symplectic form

Definition

On $V_{n} = F_{2}^{n} \oplus F_{2}^{n}$ , with coordinate order $(x, z)$ , define

$ω ((x, z), (x^{'}, z^{'})) : = x \cdot z^{'} + z \cdot x^{'} .$

This is a nondegenerate alternating bilinear form over $F_{2}$ ; $V_{n}$ has dimension $2 n$ . In characteristic two, alternating also implies symmetry.

Facts & Assumptions

Given: $n \geq 1$ , $F_{2}$ with its coordinate vector space operations, and $e_{j} \in F_{2}^{n}$ denoting the vector whose $j$ -th coordinate is $1$ and whose other coordinates are $0$ .

[F1]

Binary Pauli label and quotient: $V_{n} = F_{2}^{n} \oplus F_{2}^{n}$ with coordinatewise vector operations.

[F2]

For every prime $p$ , the two operations on $Z / p$ make it a field: $F_{2}$ is a field, so coordinate addition and multiplication satisfy the field laws.

[F3]

Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis: a basis is a linearly independent spanning family.

[F4]

Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis: dimension is the size of a finite basis.

Proof

1.1givenF2algebra

The dot product is linear in each variable, and the displayed formula is a sum of two such terms; hence $ω$ is bilinear in each of its two vector arguments.

2.1givenF2step 1.1algebra

For every $(x, z)$ , $ω ((x, z), (x, z)) = x \cdot z + z \cdot x = 2 (x \cdot z) = 0$ in $F_{2}$ , so the form is alternating. Swapping the arguments leaves its displayed value unchanged, so it is symmetric.

3.1givenstep 2.1algebra

If $x_{j} = 1$ for some $j$ , then $ω ((x, z), (0, e_{j})) = 1$ ; if $x = 0$ and $z_{j} = 1$ , then $ω ((0, z), (e_{j}, 0)) = 1$ . Thus every nonzero vector pairs nontrivially with some vector, which is nondegeneracy.

4.1givenF1F3F4algebra∎

The $2 n$ vectors $(e_{j}, 0)$ and $(0, e_{j})$ span by coordinate expansion; a zero linear combination has every coordinate zero, so they are linearly independent. By [F3] they form a basis, and [F4] gives dimension $2 n$ .

Depends on

Used by

Stabilizer encoder by symplectic elimination Algorithm
Phase-aware stabilizer tableau Definition
Stabilizer code space Definition
Stabilizer subgroup Definition
Stabilizer syndromes and logical Pauli operators Definition
A maximal stabilizer label space is self-dual Lemma
Clifford circuits lift binary symplectic transvections Lemma
Computational basis states have an extended stabilizer tableau Lemma
Phase-aware tableau updates for Clifford generators Lemma
Random Pauli measurement updates stabilizer generators Lemma
Stabilizer labels form an isotropic space of the same rank Lemma
Symplectic transvections generate the binary symplectic group Lemma
Tableau row reduction returns the deterministic Pauli sign Lemma
Clifford induces symplectic map Theorem
Clifford to binary symplectic action is surjective Theorem
CSS encoding dimension and commutation Theorem
Dimension of the logical Pauli quotient Theorem
Pauli commutator is symplectic pairing Theorem

Dependency tree · two levels

37 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Preskill, Quantum Error Correction, Physics 229 Lecture Notes, Chapter 7 (standard reference, not scraped)
M. Hein et al., Entanglement in Graph States and Its Applications (standard reference, not scraped)